Transcript Module 7 - Design of Key Rain gauge Network

RAINGUAGE
NETWORK
DESIGN
Pankaj Mani
Scientist D, National Institute of Hydrology, Patna
[email protected]
Definition
• A raingauge network is an organized system for
adequately sample the rainfall information, in
space and time, to avoid chances of over/ under
design of WR projects.
• The objective of a raingauge network, in addition
to adequately sample the rainfall, includes the
estimation of its variability within the area of
concern.
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Definition: WMO Guidelines
As per WMO guides to hydrological practices
(WMO, 1976), the network design covers
following three main aspects:
– Number of data acquisition points required,
– Location of these points and
– Duration of data acquisition from a network.
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Categories of Station
Measurement stations are divided into three
main categories, namely:
1.
2.
3.
Primary stations: Long term reliable stations expected to give
good and reliable records.
Secondary or Auxiliary stations: Placed to define the variability
over an area. The data observed at these stations are
correlated with the primary stations, and if and when consistent
correlations are obtained secondary stations can be
discontinued or removed.
Special stations: These are established for specific studies and
do not form a part of minimum network or standard network.
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Stages of Network Design
1. Data Collection
•
•
Catchment and existing raingauge information; their
location, duration, etc.
Collection of data & estimation of precipitation
characteristics,
2. Determination of accuracy of existing network,
3. Determination of number and location of new
stations required, if any.
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Methods of Network Design
1. Cv Method
2. Key Station Network Method
3. Spatial correlation Method
4. Entropy Method
5. WMO Guidelines
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Cv Method
The coefficient of spatial variation of rainfall from
the existing stations is used to determining the
optimum number of raingauges.
N = optimal number of stations,
P = allowable degree of error in the estimate of
mean rainfall and
Cv = coefficient of variation of rainfall values at
the existing m stations.
P1, P2, ........Pm is the recorded rainfall at a
known time at 1, 2, .......m station
If m > 30, σm can also be used
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Cv Method: Computational Steps
1. Use the average annual rainfall data for individual
station. (period accumulated; monthly, 10 days
average etc. can also be used)
2. Determine the statistics (mean, standard
deviation and coefficient of variation)
3. Obtain the value of P
4. Compute N (number of station) for given
permissible error in mean rainfall estimates using
formula
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Key Station Network Method
1. The station showing highest correlation with the
catchment average is assumed to be most
representative station.
2. The correlation coefficient between the average of
storm rainfall and the individual stations are
computed and the stations are arranged in
decreasing order of correlation.
3. The station exhibiting highest correlation coefficient
is called the first key station and its data is
removed for determination of next key station
.
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Key Station Network Method
1. The second and successive key stations are
determined after removing the data of already selected
key stations.
2. Now the RMSE of estimated rainfall from combination
of key stations (1,2, 3,….all) is determined and a graph
is plotted between RMSE and corresponding number of
stations in combinations.
3. It will be seen that a stage comes when the
improvement in the rainfall estimates is insignificant
with the addition of more stations.
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Key Station Network Method
Sl
No. of
Combination of stations
No. stations in
combinati
on
1
9
1,2,3,4,5,6,7,8,9
Key
station
1
2
8
2,3,4,5,6,7,8,9
4
3
7
2,3,5,6,7,8,9
8
4
6
2,3,5,6,7,9
5
5
5
2,3,6,7,9
2
6
4
3,6,7,9
9
7
3
3,6,7
7
8
2
3,6
3
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Key Station Network Method
No of
stations
RMSE
1
2
3
4
5
6
7
8
21.45 11.65 6.957 6.893 5.197 3.812 2.047 1.148
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Spatial Correlation Method



The basis of this method is the correlation function
ρ(d) exists between two stations.
The ρ(d) is a function of the distance between the
stations, and the form of which depends on the
characteristics of the area under consideration and on
the type of precipitation.
The function ρ(d) can frequently be described by the
following exponential form:
where ρ(0) is the correlation corresponding to
zero distance and d0 is the correlation radius or
distance at which the correlation is ρ(0)/e.
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Spatial Correlation Method

For an area ‘s’ with the center station, and assuming
ρ(d) exists and described as above, the variance of
the error in the average precipitation over ‘s’ is given
by Kagan (1966) as:
where, the first term is attributed
to random error and second term
with spatial variation in the
precipitation field.

The relative RMSE for an area ‘S’ with ‘N’ stations
evenly distributed so that S = N × s, is defined as:
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Spatial Correlation Method
s
s
s
s
s
L
L
s
s
s
s
s
S=N X s
L=√s
L=1.07√s
S=N X s
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SCM:Computational Steps

Theoretically, ρ(0) should equal to unity but is rarely
found so in the practice due to random errors in
precipitation measurement and micro climatic
irregularities over an area.

The correlation between two rainfall stations with
concurrent data are computed and plotted against its
distance. For ‘n’ number of existing stations the
possible combination would be nC2. The inter-stations
distance is plotted on X-axis while the correlation is
plotted on Y-axis (on log scale).
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SCM:Computational Steps
Station Name
6
4
2
2
2
5
4
10
5
7
5
2
3
10
2
3
10
2
10
9
9
10
1
1
3
6
1
6
6
5
4
8
1
10
6
3
4
1
1
5
4
8
7
8
Distance
6.635933
6.720109
7.234885
8.698046
9.633291
10.21089
10.71486
12.02968
13.37846
13.61854
14.43904
14.46311
14.61809
15.72948
16.2681
16.85225
17.21371
18.79007
20.59654
20.63936
20.87076
23.24876
Correlation
0.281523
0.487851
0.448008
0.280573
0.480566
0.17029
0.346644
0.254124
0.355068
0.368347
0.296527
0.645556
0.24556
0.476863
0.427164
0.382571
0.451507
0.25819
0.468624
0.396006
0.281303
0.343275
Station Name
3
3
5
5
6
7
3
2
2
3
1
4
7
10
4
9
9
9
2
9
9
4
5
8
7
8
6
7
7
8
8
8
8
1
9
7
5
6
3
9
1
4
Distance
23.42803
23.76972
25.3396
31.36899
31.99795
34.21938
34.91605
36.98174
37.68367
38.2408
38.24285
38.67579
40.85275
42.0461
43.41789
45.93341
51.81685
55.24055
56.30718
58.28824
59.21265
Correlation
0.415388
0.24444
0.281992
0.214145
0.299989
0.907909
0.257129
0.282765
0.422015
0.369154
0.399103
0.39844
0.298737
0.248119
0.335313
0.193902
0.265757
0.177415
0.430246
0.36881
0.335097
SCM:Computational Steps
The slope of this line will 1/d0
and the Y intercept will give
log[ρ(0)], hence ρ(0) and d0 is
computed.
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SCM:Computational Steps
• S= total catchment area, n = no. of station evenly
distributed representing area ‘s’ by each station, so
that S=n x s.
• Compute the sigma and mean for each station from
time series rainfall data, estimate Cv = sigma/mean.
Compute s=S/n
• Compute RMSE, Z1 (for estimation average rainfall
over area S) for n=1, 2, 3, ….n
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SCM:Computational Steps
No. of Stations
RMSE
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.776178
0.388089
0.258726
0.194045
0.155236
0.129363
0.110883
0.097022
0.086242
0.077618
0.070562
0.064682
0.059706
0.055441
0.051745
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Entropy Method
1. The hydrological information and regional
uncertainty associated with a set of
precipitation station are estimated using
Shannon’s entropy concept.
2. Since time series of rainfall data can be
represented by gamma distribution due to the
presence of skewness, the entropy term is
derived using single and bivariate gamma
distribution.
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Entropy Method
• A quantitative measure of the uncertainty associated
with a probability distribution or the information content
of the distributions termed Shannon entropy can be
mathematically expressed as:
where H(X) is the entropy corresponding
to the random variable X; k is a constant
that has value equal to one, when
natural logarithm is taken; and i p
represents the probability of ith event of
random variable X.
•
Marginal entropy for the discrete random variable X is
defined as:
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Entropy Method
•
Marginal entropy for the discrete random variable X is
defined as:
where p(Xi ) is the probability of
occurrence of Xi, computed by N2 and
LN2 distributions, and N is the number of
observations.
•
•
The marginal entropy H(X) indicates the amount of
information or uncertainty that X has.
If the variables X and Y are considered as independent,
then the joint entropy [H(X, Y)] is equal to the sum of
their marginal entropies defined by:
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Entropy Method
• Similarly, the joint entropy in a region with m station and
precipitation variables (X1, X2, ……Xm) can be extended
to
where, X1, X2, ……. Xm represents
precipitation variables measured at m
station and p(xj1, xj2, ….xjm) is the joint
probability of occurrence of jth event at mth
station.
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Entropy Method
• It means, the entropy H(Xi) at a station,
which is also the transmitted information
about the variable Xi at station I can be
decomposed into:
a) the net information H(Xi) at i,
b) common information T(Xi, Xj) between station
pair i and j which will remain constant
throughout the segment i-j considering station
pair (i,j), which depends on the the distance
between the station pair and is maximum at
station location
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Entropy Method
z and w are normalized variates
of X and Y respectively, with a
mean of zero and standard
deviation of unity. If ρzw is the
correlation coefficient between z
and w, then the information
transmitted by variable Y about
X, T(X,Y) or by variable X about
Y, T(Y,X) is given by Shannon
and Weaver
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EM: Computational Steps
• Using the rainfall time series data, the mean and
variance for each station are estimated. Using these
values compute the scale and shape parameters (σ and
λ) for gamma distribution; mean = σ λ and variance = σ2
λ.
• The entropy of rainfall variable measured at each station,
H(X) were computed using formula;
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EM: Computational Steps
• Normalize the rainfall distribution at a station using its
mean and sigma value (use x-sigma/mean).
• Make the possible pair of inter-station by making
triangular grids from existing rainfall stations such that:
– correlation between all three stations at the vertices exist, and
– none of triangles has any obtuse angle.
• Now compute the correlation coefficient (ρzw ) between
all pair of stations using their normalized series.
Compute T(X,Y)= T(Y,X) by formula:
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EM: Computational Steps
• Considering a triangular element shown below formed by
joining three precipitation station (i,j,k) measuring
precipitation Xi, Xj, Xk respectively.
K
D
1
H (X i )-T (X X j )
i
H (X )
i
D2
H (X j )-T (X i X j)
H (X i )
T (X X k)
i
K'
I
d1
D
J
d2
T (X X )
j k
T (X i X j )
I
K
J
• Interpolate H(XiXj) at intermediate points (points at 10
equal interval spacing between two stations) considering
the figures and assuming linear transformation of
information between two stations.
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EM: Computational Steps
• Now
plot
the
contours
using
H(XiXj).
information
computed
• Identify
the
pockets
inadequate
information
propose the new stations.
of
to
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Network Design: WMO Guidelines
These recommendations are based on the 1991 review of Members’ responses
regarding the WMO basic network assessment project
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Network Design: WMO Guidelines
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